Using an amplitude analysis to measure the photon polarisation in $B \to K\pi\pi\gamma$ decays

A method is proposed to measure the photon polarisation parameter $\lambda_\gamma$ in $b\to s\gamma$ transitions using an amplitude analysis of $B\to K\pi\pi\gamma$ decays. Simplified models of the $K\pi\pi$ i system are used to simulate $B^+ \to K^+\pi^-\pi^+\gamma$ and $B^0\to K^+\pi^-\pi^0$ decays, validate the amplitude analysis method, and demonstrate the feasibility of a measurement of the $\lambda_\gamma$ parameter irrespective of the model parameters. Similar sensitivities to $\lambda_\gamma$ are obtained with both the charged and neutral hadronic systems. In the absence of any background and distortion due to experimental effects, the statistical uncertainty expected from an analysis of $B^+ \to K^+\pi^-\pi^+\gamma$ decays in an LHCb data set corresponding to an integrated luminosity of $9\,$fb$^{-1}$ is estimated to be $0.009$. A similar measurement using $B^0\to K^+\pi^-\pi^0$ decays in a Belle II data sample corresponding to an integrated luminosity of $5\,$ab$^{-1}$ would lead to a statistical uncertainty of $0.018$